The Master is evil, but not in a mindless way like many of the Doctor's other enemies. He is extremely smart and cunning. He enjoys causing chaos and takes pleasure in having the Doctor as a nemesis. He desires power and is consumed with a desire to destroy the Doctor, who imrisoned him temporarily on Earth (Terror of the Autons, 1971) and has thwarted many of his schemes. The Master's favored weapon is a short, black rod called the Tissue Compression Eliminator, which reduces a person into a tiny, shrunken corpse. He has also been known to use the inherent mental powers of the Time Lords in order to control the minds of others, forcing them to do his bidding. His TARDIS (unlike that of the Doctor) is usually fully functional (including its chameleon circuit, a component with which the Doctor has had extensive problems), and it is even a more recent model than the one used by the Doctor.

The Master is one of only a few Time Lords the Doctor has faced, and is by far the Time Lord he has most frequently encountered. A few of the others are:

The Master returned as a primary nemesis for The Doctor in the revival of Doctor Who. So far, he appeared in three episodes of Series 3, and returned for the climatic special "The End of Time". Whether he survived that episode is not clear, but given the long run of the character on Doctor Who, it is likely that he will return at some point.

In the narrative established in the story, The Master fled from the cataclysmic conflict known as The Last Great Time War, hiding himself at the limits of time, in the year 100 Trillion. He physically and mentally gives himself the identity of "Professor Yana" (played by Derek Jacobi), a kind and intelligent man, and forgets that he is The Master. It is only when The Tenth Doctor visits that he comes back to his senses, and after being killed, regenerates as a younger looking man, played by John Simm. He then attempts acts of megalomania and tyranny that are ingenious and devious, but are still thwarted by The Doctor.

I wish I was more conversant with Classic Doctor Who, but while the Master has always been a genius and a schemer, the latest regeneration of The Master is also insane, although just as intelligent as ever. His tyranny breaks into sadism and mind games that go beyond normal cruelty. He becomes something of a Caligula or a Nero, a jaded yet almost childlike figure, gleefully watching The Teletubbies before ordering killings.

One of the questions about the possible return of The Master is whether he will be played with John Simm, with the same insanity, or whether he will be played by another actor, with another take on The Master's role as the devious tyrant.